for \(i \in [n]\) , one has that \(\left|\left\langle z^* , z^{\prime } \right\rangle  \right|\)  is large which is
enough to address the first item. More precisely implicit
in [1]}, for any constant \(\beta \in (0,1)\)  as long as
parity sampling is sufficiently strong we have
\({\mathbb {E}}_{z \sim {\mathbf {Z}^{\otimes }\vert _{(S,\sigma )}}}\left[\left\langle z^{\prime } , z \right\rangle ^2 \right] \ge 1 - \beta .\) 
